Comparison Test (series)

Comparison Test: Let $0\le a_n\le b_n$ for all sufficiently large $n$.

• If $\sum_{n=1}^\infty b_n$ converges , then $\sum_{n=1}^\infty a_n$ converges.
• If $\sum_{n=1}^\infty a_n$ diverges , then $\sum_{n=1}^\infty b_n$ diverges.

This test reduces convergence questions to bounding terms by simpler expressions.

Proof sketch (optional): Use monotonicity of partial sums for nonnegative series: $\sum_{k=1}^N a_k \le \sum_{k=1}^N b_k$. Boundedness of the larger partial sums forces boundedness (hence convergence) of the smaller ones.